In mathematics, specifically algebraic topology, an Eilenberg–MacLane space is a topological space with a single nontrivial homotopy group. Let G be a...

algebraic topology, there is a distinguished class of spectra called Eilenberg–MacLane spectra H A {\displaystyle HA} for any abelian group A {\displaystyle...

corresponding spectrum H A {\displaystyle HA} has n {\displaystyle n} -th space K ( A , n ) {\displaystyle K(A,n)} ; it is called the Eilenberg–MacLane spectrum of...

variety as a sum of sheaf cohomology groups. Spectrum: H Z p {\displaystyle HZ_{p}} (Eilenberg–Maclane spectrum of the integers mod p {\displaystyle p} .)...

products over the sphere spectrum by tensor products over Z {\displaystyle \mathbb {Z} } (or the Eilenberg–MacLane-spectrum H Z {\displaystyle H\mathbb...

and whose other homotopy groups are zero. Such a space is called an Eilenberg–MacLane space. This space has the remarkable property that it is a classifying...

{\begin{matrix}X\\\downarrow \\K\end{matrix}}} where K {\displaystyle K} is an Eilenberg–Maclane spectrum representing the generators of H ∗ ( X ) {\displaystyle H^{*}(X)}...

functor known as the topological Hochschild homology of the ring's Eilenberg–MacLane spectrum. If R is a constant simplicial ring, then this is the same thing...

space Cohomotopy Cohomotopy group Brown's representability theorem Eilenberg–MacLane space Fibre bundle Möbius strip Line bundle Canonical line bundle...

should be observed if the Eilenberg–Maclane spectrum H F p {\displaystyle H\mathbb {F} _{p}} is replaced by an arbitrary spectrum E {\displaystyle E} , then...

given by the ring spectrum H Q {\displaystyle H\mathbb {Q} } , called the Eilenberg–Maclane spectrum, constructed from the Eilenberg–MacLane spaces K ( Q ...

respectively. Taking homology and cohomology with respect to an Eilenberg–MacLane spectrum recovers Alexander duality formally. Spanier, Edwin H.; Whitehead...

}(k)}(X_{+},K(p,q,A))=H^{p,q}(X,A)} showing these sheaves represent motivic Eilenberg-Maclane spacespg 3. A further construction in A1-homotopy theory is the category...

{\displaystyle \phi _{n}:X_{n}\to X_{n-1}} that are fibrations with fibers Eilenberg-MacLane spaces K ( π n ( X ) , n ) {\displaystyle K(\pi _{n}(X),n)} . In short...

n)]=\operatorname {H} ^{n}(X;A)} where K ( A , n ) {\displaystyle K(A,n)} is the Eilenberg–MacLane space. The above equation leads to the notion of a generalized cohomology...

t}\to E_{r}^{s-1,t+r}} . Some of the simplest calculations are with Eilenberg–Maclane spectra such as X = H Z {\displaystyle X=H\mathbb {Z} } and X = H...

{\displaystyle H^{n}(X,G)}  of the space X. One says that the omega-spectrum of Eilenberg-MacLane spaces are representing spaces for singular cohomology with...

S^{n+m}} . It is also not hard to write down multiplication maps for Eilenberg-MacLane spectra (representing ordinary cohomology) and certain Thom spectra...

data. Samuel Eilenberg, Polish-American mathematician, Eilenberg–MacLane space, Eilenberg–Mazur swindle, Eilenberg–Maclane spectrum, Eilenberg–Steenrod axioms...

Module spectrum Category of graded vector spaces Category of representations Change of rings Morita equivalence Stable module category Eilenberg–Watts...

algebras A / F p {\displaystyle A/\mathbb {F} _{p}} Recall that the Eilenberg–MacLane spectrum can be embed ring objects in the derived category of the integers...

Eckmann–Hilton argument. Eckmann–Hilton duality Eilenberg–MacLane spaces Given an abelian group π, the Eilenberg–MacLane spaces K ( π , n ) {\displaystyle K(\pi...

periodicity Eilenberg–MacLane space Free loop Fundamental group Gray's conjecture List of topologies Loop group Path (topology) Quasigroup Spectrum (topology)...

space having G as its fundamental group, namely the corresponding Eilenberg–MacLane space. Thus, the group cohomology of Z {\displaystyle \mathbb {Z}...

cobordism is a product of Eilenberg–MacLane spectra – MO = H(π∗(MO)) – while oriented cobordism is a product of Eilenberg–MacLane spectra rationally, and...

classifying space of the category G L ( R ) {\displaystyle GL(R)} or the Eilenberg–MacLane space of the type K ( G L ( R ) , 1 ) {\displaystyle K(GL(R),1)} ...

group and then killing (eliminating) it with a fibration involving an Eilenberg–MacLane space. In principle this gives an effective algorithm for computing...

two normal matrices. Eilenberg–Ganea conjecture: a group with cohomological dimension 2 also has a 2-dimensional Eilenberg–MacLane space K ( G , 1 ) {\displaystyle...

currents, Annals of Mathematics, volume 72 (1960), pp. 458–520. 1987 Samuel Eilenberg for his fundamental contributions to topology and algebra, in particular...

Conformal space Complex analytic space Drinfeld's symmetric space Eilenberg–Mac Lane space Euclidean space Fiber space Finsler space First-countable space...